A mattress cover sits between your mattress and your bedsheet, and its whole job is to take the wear that would otherwise land directly on the mattress - spills, dust, sweat, and the slow flattening that comes from years of direct use. Which cover to buy really comes down to one question: do you need it to be waterproof, or just breathable?
A PVC cover is the one to reach for if there's a real chance of liquid getting to the mattress - a child's bed, a guest room you don't want to worry about, or simply peace of mind against accidental spills. It won't let anything through, but it also won't breathe the way fabric does, so some people prefer to only use it on the top mattress layer and pair it with a regular cotton bedsheet over it.
A cotton checkered cover is the better everyday choice when your priority is comfort and airflow rather than spill-proofing - it's quilted, sits like a fitted sheet, and washes in a machine like any other bedding.
Measure your mattress before ordering rather than going by "single" or "double" alone - mattress heights vary, and a cover sized for a thinner mattress can feel tight on a thicker one.
PVC covers just need a wipe-down with a damp cloth - no washing machine, no drying. Cotton covers are machine washable; wash separately for the first couple of washes since checkered dyes can run slightly, and avoid high heat when drying to keep the quilting from puckering.